When it comes to tearing down your tailgating site, apply the camping rule that says you should take out everything you bring in. You might be able to find some trash cans nearby, but then again, you might not. It's important to make sure you have everything with you to properly dispose of, or carry away, all of your trash.
If you grill out using a charcoal grill, keep in mind the policies and rules for your tailgating area. Charcoal coals can stay hot for hours and they need to be disposed of properly. Using aluminum pans that you had the food in is a good way to discard the remaining charcoal. You can then use the melted ice in the cooler to pour over the coals if they're still hot.
Whatever you do, don't leave trash, food or bottles laying around when you're done. It's in everyone's best interest to keep the place nice and neat so you'll have a premium spot to tailgate throughout the rest of the season.